A key problem in AI is to develop intelligent systems and services that
actively gather most relevant information. In recent years, a
fundamental problem structure has emerged as extremely useful for
addressing this problem: Submodularity is an
intuitive diminishing returns property, stating that adding an element
to a smaller set helps more than adding it to a larger set.
Applications where this property is useful include active
learning and experimental design, informative
path planning, multi-agent coordination, structure learning,
clustering, influence maximization, weblog ranking, trading off utility
and privacy as well as game theoretic applications. In contrast to most
existing approaches, submodularity allows to efficiently find provably
(near-)optimal solutions.
In this tutorial, we will give an introduction to the concept of
submodularity, discuss algorithms for optimizing submodular
functions and illustrate their usefulness in solving difficult AI
problems, with a special focus on active information gathering
tasks. Since submodularity arises in so many different areas of AI, and
since information gathering is central to many AI
applications, we believe that this property is both of theoretical and
practical interest to a large part of the AI community. This
tutorial will not require prior knowledge beyond the basic concepts
covered in an introductory AI class.

The main objective of this tutorial is to introduce the concept of
submodular function optimization and its emerging importance in solving
complex AI problems. As a special focus, we illustrate the concept on a
key AI task: Intelligent gathering of most relevant information, in a
variety of complex real-world problems.
Since submodularity arises in so many different areas of AI, and since
information gathering is central to many AI applications, we believe
that this property is both of theoretical and practical interest to a
large part of the AI community..

Andreas Krause is
an assistant professor of Computer Science at the California Institute
of Technology. He received his Ph.D. from Carnegie Mellon
University in 2008. He is a recipient of a Microsoft Research Graduate
Fellowship, and his research on sensor placement and information
acquisition received awards at several major conferences (KDD '07, IPSN
'06, ICML '05 and UAI '05) and the ASCE journal of Water Resource
Planning and Management.

Carlos Guestrin is
the Finmeccanica Assistant Professor in the Machine Learning and
Computer Science Departments at Carnegie Mellon University. Previously,
he was a senior researcher at Intel, and received his PhD from Stanford
University. Carlos' work received awards at a number of major
conferences and a journal. He is also a recipient of the ONR Young
Investigator Award, the NSF Career Award, the Alfred P. Sloan
Fellowship, the IBM Faculty Fellowship, and was named one of the 2008
Brilliant 10 by Popular Science Magazine. Carlos is currently a member
of the Information Sciences and Technology (ISAT) advisory group for
DARPA.

[8] A. K. Kelmans and B. N. Kimelfeld. Multiplicative
submodularity of a matrix’s principal minor as a function of
the set of its rows and some combinatorial applications. Discrete
Mathematics, 44(1):113–116, 1980.

Proves that entropy is a submodular function.

[9] D. Kempe, J. Kleinberg, and E. Tardos. Maximizing the
spread of influence through a social network. In KDD, 2003 [pdf]

Maximizing influence (viral marketing) in social networks
is submodular.